This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).

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Definition of truth in first-order logic

Let $L$ be a first order language. Let $P$ be a predicate symbol from $L$, and $c$ a constant. Given an interpretation $I$ of $L$, a definition states The formula $P(c)$ is true in $I$ iff $c\in ...
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2answers
83 views

What is the exact role of logic in the foundations of mathematics?

At high school and in the beginning of my university studies, I used to believe the following "equation": Foundations of mathematics = Logic + Set Theory Of course, this "equation" does not hold ...
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0answers
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Any “exotic” foundations of mathematics? [closed]

There is a myriad of axiomatizations of set theory (a branch of mathematics obviously not at all identical with notorious ZFC) and other formal systems working with classes, categories and such. All ...
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1answer
62 views

Are the 1 dimensional loop spaces in homotopy type theory commutative?

Theorem 2.1.6 of the Homotopy Type Theory book proves that $\Omega^{2}A$ is always commutative, using a similar argument to the one used for loop spaces in algebraic topology. Isn't it also the case (...
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1answer
51 views

If a theory $A$ can prove $B$ consistent relative to $A$, and $A$ is consistent, does $B$ have to be consistent?

Let's say we have two sets of axioms $A$ and $B$ such that $\mathsf{ZF} \subseteq A \subseteq B$, and from $A$ we can prove that if $A$ is consistent, then $B$ is consistent as well (that is, $A \...
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4answers
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Switch algebraic sign

I can't believe that I seriously ask this question as it is so simple. Given this $-x^3+4x$ I'd like to factor out -x, so I did $-x(x^2-4)$ which equals $-x(x^2-2^2)$ equals $-x(x-2)(x+2)$ ...
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0answers
53 views

Is there a schsim between classical logic and categorical logic?

I've been trying to learn a little bit more about the foundations of mathematics, and it has strike me that there seems to be two competing points of view about what the foundations should be. While ...
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3answers
243 views

Preserving equality between different mathematical objects

I'm taking an 'Intro to Higher Mathematics'-type course right now, were we learn about basic set theory, number theory, algebra, etc. and I had the following thought: Say you're trying to solve a ...
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0answers
67 views

Could mathematical reasoning be non-axiomatic?

"Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, ...
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1answer
85 views

Existential axioms for category theory

There are some existential axioms in set theory, for example, axiom schema of specification. It's my understanding that category theory isn't based essentially on set theoretic foundation. If so, I ...
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2answers
1k views

Is there a model of ZFC inside which ZFC does not have a model?

Assuming ZFC has a model, is there a model of ZFC such that in that model, ZFC has no model? Also, assuming ZFC has a model, is there a model of ZFC such that in that model, ZFC is inconsistent?
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0answers
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The monadic second order theory with $<$ and Presburger arithmetic

Consider the monadic second order logic over the natural numbers with $<$ as a predicate, i.e. the second order logic over $(\mathbb N, 1, <)$, where we can quantify over sets and individual ...
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2answers
85 views

Formal systems in which $\forall x \in \mathbb{R}(x \neq 0 \rightarrow x^{-1} \neq 0)$ is true, but the contrapositive is disallowed.

Question. Are there any formal systems out there for which $$\forall x \in \mathbb{R}(x \neq 0 \rightarrow x^{-1} \neq 0)$$ is true, but the contrapositive $$\forall x \in \mathbb{R}(x^{-1} =...
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0answers
22 views

Symmetry vs. commutativity and more..

I was reading the following segment of an article about commutativity: ...
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1answer
55 views

Axiom in Foundations, Extensionality

In my Foundations of Mathematics Textbook I encountered the following problem. The book states that for the domain of discourse $D = \{a,b,c\}$ and binary relation defined as $E = \{(a,b),\, (a,c)\}$ ...
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2answers
786 views

Is TOP a small category?

A quick question... is the category of topological spaces and continuous maps a small category? If so how do we know and if not how do we know not?
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2answers
200 views

How many sorts are there in Terry Tao's set theory?

In his 2010 post, A computational perspective on set theory, Terry Tao writes: The standard modern foundation of mathematics is constructed using set theory. With these foundations, the ...
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1answer
98 views

Are sets and symbols the building blocks of mathematics?

A formal language is defined as a set of strings of symbols. I want to know that if "symbol" is a primitive notion in mathematics i.e we don't define what a symbol is. If it is the case that in ...
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1answer
96 views

Is it circular to define the Von Neumann universe using “sets”?

I was just reading the Wikipedia page on the Von Neumann universe, where it is stated that this universe "is often used to provide an interpretation or motivation of the axioms of ZFC." However, later ...
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2answers
92 views

Explicit example of countable transitive model of $\sf ZF$

Do we know any explicit example of a countable transitive model for $\sf ZF$ or $\sf ZFC$?
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1answer
82 views

Why wasn't Bertrand Russell surprised by the set of all sets that contain themselves?

Russell's paradox deals with the question: "Does the set of all sets that do not contain themselves, contain itself?" What about the question: "Does the set of all sets that contain themselves, ...
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0answers
28 views

Definition of Operation

What are operations in Mathematics? I do not find it formally defined anywhere. What is the difference between operation and function? Earlier I thoght operations are just binary operations. But later ...
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1answer
42 views

Why do metric spaces that produce the same topology have different number theoretical difficulties?

Consider finding a a point with rational distance to the corners of unit square. Under the Euclidean metric this is very hard. (unsolved) Under the "city block" or taxicab metric this is very easy ...
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1answer
29 views

Degenerate zeroes, fundamental theorem of algebra.

The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two ...
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1answer
119 views

Does PA prove that Con(PA) implies Con(ZF-I) and Con(NFU)?

I read from many sources that PA and ZF-I (a suitable axiomatization of ZF minus Infinity plus its negation) are bi-interpretable, but is PA enough to prove that they are equiconsistent? Specifically ...
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How to Prove? - If the interpretation of theory is consistent, then the interpreted theory is consistent

Let L1 and L2 be finite or recursive languages, and T a theory in the language of L2. A translation of L1 into T is an assignment to each sentence S of L1 into a sentence i(S) of L2 such that: (T0) i(...
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1answer
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Justification of ZFC without using Con(ZFC)?

I saw this post by Carl Mummert, which describes how one can 'see' the reasonableness of ZFC, via iterating the power-set operation starting from the empty set. However, this iterations proceed in ...
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4answers
2k views

Are professional mathematicians concerned with formalizing infinitely many dependent choices?

I've noticed certain arguments in analysis textbooks which rely on the principle of being able to pick elements infinitely many times. For example, an argument might go "Pick $x_1\in S$ such that $P(...
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1answer
99 views

How do you visualize $\mathcal{P}(1)$ in constructive mathematics?

If I understand correctly, constructive mathematics doesn't prove that the powerset $\mathcal{P}(X)$ of a set $X$ is a Boolean algebra; in general, all we can say is that its a Heyting algebra. This ...
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1answer
44 views

Are there other methods of proof other than contrapositive, induction, contradiction, construction, and counter example?

I have only heard of a few methods of proof, namely, contrapositive, induction, contradiction, construction, and counter example. Are there other types of proofs?
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9answers
2k views

Is formal truth in mathematical logic a generalization of everyday, intuitive truth?

I'm trying to wrap my head around the relationship between truth in formal logic, as the value a formal expression can take on, as opposed to commonplace notions of truth. Personal background: When I ...
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0answers
71 views

Are all statements about math inherently formal? Can one do math without formal logic? [duplicate]

Are all people who do mathematics applying (whether they know it or not) formal logic? Does every statement someone may make about math, at its core, a formal statement in mathematical logic? (I'm ...
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1answer
75 views

Existence of some categories in the category of categories

I'm studying category of categories. I read that when there are categories $A,B$, it is allowed to define the product $A\times B$. Equalizers and coequalizers also exist. However, there are some ...
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1answer
78 views

Has it been proved that ∨ ∧ ¬ ⇒ ⇔ ∀ ∃ and predicates are enough to express all known mathematics?

I frequently encounter this statement in math books: "All of known mathematics can be expressed in terms of elementary predicates, logical connectives, and quantifiers."
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1answer
47 views

Different Mathematics

Hey I am a high school student who is very interested in the philosophy of mathematics. I was watching this talk by Stephen Wolfram about whether or not mathematics is invented or discovered. In it he ...
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3answers
160 views

In which order should I learn the foundations of mathematics? [closed]

I know from Wikipedia that those are the four pillars of the foundations of mathematics: Proof theory Aximatic Set theory Model Theory Recursion Theory and I want to learn all of them, the problem ...
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1answer
111 views

Can these definitions of the words “problem” and “solution” be formalized, and if so, has this been done? If so, where can I learn more about it?

I had a thought. Define that: Vague Definition 0. A problem consists of: a set $X$ a set $Y$ a function $f : X \rightarrow Y$ a way $\overline{X}$ of representing the elements of $X$ ...
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1answer
129 views

How can mathematics work in wildly different set theories?

There are several set theories, e.g. ZFC and NF, which often have different axioms or are even outright contradictory. And yet most of other branches of mathematics, e.g. abstract algebra or topology, ...
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45 views

Is PA the most common foundation for arithmetic?

Are Peano Axioms the most common and widely accepted axiomatization of arithmetic, just as ZFC is the most common and widely accepted foundation for all of mathematics?
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0answers
43 views

What type of reasoning is employed when studying metalogic?

When studying, or doing any kind of reasoning about logic, what type of logic is used? Or how is the reasoning structured?
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2answers
73 views

How are image and pre-image different from range and domain respectively?

How are image and pre-image different from range and domain respectively, in Layman's terms (as simple as possible)? Are they basically just keywords that often indicate more nuanced subsets of the ...
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2answers
392 views

Is PA the first axiomatization of arithmetic to be discovered? [closed]

Is Peano Arithmetic the first axiomatization of arithmetic to be discovered?
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0answers
27 views

Why is it impossible to prove absolute consistency of a theory falling prey to Godel's theorems?

Why is it impossible to prove absolute consistency of a theory T falling prey to Godel's theorems? I understand that a theory falling prey to Godel's second incompleteness theorem cannot prove its own ...
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1answer
23 views

Need an assistance with a specific step of a specific Division Algorithm proof

I'm trying to wrap my head around a Division Algorithm's proof. That is, Let $a, b \in \mathbb{Z}, a \neq 0$. Then there are unique $q,r \in \mathbb{Z}$ such that $b = qa + r, 0 \leq r < |a|$. ...
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2answers
828 views

Why doesn't this definition of natural numbers hold up in axiomatic set theory?

I was reading about older definitions of the natural numbers on Wikipedia here (in retrospect, not the best place to learn mathematics) and came across the following definition for the natural numbers:...
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2answers
56 views

Impossibility of proving a foundation to be consistent

An argument came to my mind that seems really mind-blowing and I haven't found it anywhere. Here's how it goes: We call a formal system F embodied in classical logic a foundation of mathematics when ...
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1answer
47 views

Why include equality in FOL for ZFC?

What are the pros and cons of working with first-order logic with equality for constructing ZFC, when all you have to do is make '$x=y$' a shorthand for: $$'\forall z [z \in x \Leftrightarrow z \in y] ...
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1answer
36 views

Any foundational theory of math falls prey to the incompleteness theorems - true or false?

I heard somewhere on the internet once something along the following lines: Any conceivable foundational theory of mathematics (be it ZFC or, if ZFC was found to be inconsistent, some modification ...
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1answer
60 views

Is isomorphism defined between large categories?

By definition, an isomorphism between two objects in a category is a morphism so and so.. We know that $\mathbf {Cat}$ is the categories of small categories so that morphisms between objects are ...
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1answer
39 views

Could relational operators be used to construct formal theory of natural numbers which is “stronger” than Peano Axioms?

This is a beginner's question about foundational construction of (alternative?) number theory. The notion of mathematical equality is closely related to logico-philosophical notion of 'Law of Identity'...